##
**Functional equations related to inner product spaces.**
*(English)*
Zbl 1177.39041

Summary: Let \(V,W\) be real vector spaces. It is shown that an odd mapping \(f:V\to W\) satisfies

\[ \sum^{2n}_{i=1}f(x_i-1/2n\sum_{j=1}^{2n}x_j)=\sum_{i=1}^{2n}f(x_i)-2_nf(1/2n\sum_{i=1}^{2n}x_i) \]

for all \(x_1,\dots,x_{2n}\in V\) if and only if the odd mapping \(f:V\to W\) is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

\[ \sum^{2n}_{i=1}f(x_i-1/2n\sum_{j=1}^{2n}x_j)=\sum_{i=1}^{2n}f(x_i)-2_nf(1/2n\sum_{i=1}^{2n}x_i) \]

for all \(x_1,\dots,x_{2n}\in V\) if and only if the odd mapping \(f:V\to W\) is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

### MSC:

39B82 | Stability, separation, extension, and related topics for functional equations |

39B52 | Functional equations for functions with more general domains and/or ranges |

### Keywords:

inner product spaces; Cauchy additive; Hyers-Ulam stability; functional equation; Banach spaces
PDF
BibTeX
XML
Cite

\textit{C. Park} et al., Abstr. Appl. Anal. 2009, Article ID 907121, 11 p. (2009; Zbl 1177.39041)

### References:

[1] | S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1960. · Zbl 0086.24101 |

[2] | D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222-224, 1941. · Zbl 0061.26403 |

[3] | T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64-66, 1950. · Zbl 0040.35501 |

[4] | Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297-300, 1978. · Zbl 0398.47040 |

[5] | P. G\uavruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431-436, 1994. · Zbl 0818.46043 |

[6] | F. Skof, “Proprietà locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113-129, 1983. |

[7] | P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76-86, 1984. · Zbl 0549.39006 |

[8] | S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der UniversitHamburg, vol. 62, pp. 59-64, 1992. · Zbl 0779.39003 |

[9] | J. M. Rassias, “On the stability of the Euler-Lagrange functional equation,” Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185-190, 1992. · Zbl 0789.46036 |

[10] | P. G\uavruta, “On the Hyers-Ulam-Rassias stability of the quadratic mappings,” Nonlinear Functional Analysis and Applications, vol. 9, no. 3, pp. 415-428, 2004. · Zbl 1066.39029 |

[11] | D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. · Zbl 0907.39025 |

[12] | Th. M. Rassias, “New characterizations of inner product spaces,” Bulletin des Sciences Mathétiques, vol. 108, no. 1, pp. 95-99, 1984. · Zbl 0544.46016 |

[13] | M. S. Moslehian and F. Zhang, “An operator equality involving a continuous field of operators and its norm inequalities,” Linear Algebra and Its Applications, vol. 429, no. 8-9, pp. 2159-2167, 2008. · Zbl 1156.47021 |

[14] | M. S. Moslehian, K. Nikodem, and D. Popa, “Asymptotic aspect of the quadratic functional equation in multi-normed spaces,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 717-724, 2009. · Zbl 1168.39012 |

[15] | B. Bouikhalene and E. Elqorachi, “Ulam-G\uavruta-Rassias stability of the Pexider functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 27-39, 2007. · Zbl 1130.39022 |

[16] | P. G\uavruta, “An answer to a question of John M. Rassias concerning the stability of Cauchy equation,” in Advances in Equations and Inequalities, Hardronic Mathical Series, pp. 67-71, Hadronic Press, Palm Harbor, Fla, USA, 1999. |

[17] | P. G\uavruta, M. Hossu, D. Popescu, and C. C\uapr\uau, “On the stability of mappings and an answer to a problem of Th. M. Rassias,” Annales Math!tiques Blaise Pascal, vol. 2, no. 2, pp. 55-60, 1995. · Zbl 0853.46036 |

[18] | K.-W. Jun, H.-M. Kim, and J. M. Rassias, “Extended Hyers-Ulam stability for Cauchy-Jensen mappings,” Journal of Difference Equations and Applications, vol. 13, no. 12, pp. 1139-1153, 2007. · Zbl 1135.39013 |

[19] | K.-W. Jun and J. Roh, “On the stability of Cauchy additive mappings,” Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 15, no. 3, pp. 391-402, 2008. · Zbl 1156.39018 |

[20] | D. Kobal and P. Semrl, “Generalized Cauchy functional equation and characterizations of inner product spaces,” Aequationes Mathematicae, vol. 43, no. 2-3, pp. 183-190, 1992. · Zbl 0755.39007 |

[21] | Y.-S. Lee and S.-Y. Chung, “Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions,” Applied Mathematics Letters of Rapid Publication, vol. 21, no. 7, pp. 694-700, 2008. · Zbl 1152.39318 |

[22] | P. Nakmahachalasint, “On the generalized Ulam-G\uavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 63239, 10 pages, 2007. · Zbl 1148.39026 |

[23] | C.-G. Park, “Stability of an Euler-Lagrange-Rassias type additive mapping,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 101-111, 2007. |

[24] | A. Pietrzyk, “Stability of the Euler-Lagrange-Rassias functional equation,” Demonstratio Mathematica, vol. 39, no. 3, pp. 523-530, 2006. · Zbl 1113.39034 |

[25] | C.-G. Park and J. M. Rassias, “Hyers-Ulam stability of an Euler-Lagrange type additive mapping,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 112-125, 2007. |

[26] | K. Ravi, M. Arunkumar, and J. M. Rassias, “Ulam stability for the orthogonally general Euler-Lagrange type functional equation,” International Journal of Mathematics and Statistics, vol. 3, no. A08, pp. 36-46, 2008. · Zbl 1144.39029 |

[27] | K. Ravi and M. Arunkumar, “On the Ulam-G\uavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation,” International Journal of Applied Mathematics & Statistics, vol. 7, pp. 143-156, 2007. |

[28] | C. Park, J. Lee, and D. Shin, “Quadratic mappings associated with inner product spaces,” preprint. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.